The present invention relates to the compression of antenna voltage data acquired with a receiving element of an antenna system. More particularly, the present invention relates to a data compression method for determining a set of complex weights which, in cooperation with the position and orientation of one or more idealized linear dipole antennas, permit voltage data measured with a receiving antenna to be represented with minimal loss of accuracy using minimal computer storage memory.
In general, data compression refers to the process of reducing the amount of data required to represent a given quantity of information with sufficient practical accuracy. A practical problem solved by data compression is efficient image transmission where the underlying objective is to remove redundant data. The reduced data may thereby permit more efficient manipulation, storage, and transmission of information. At a later stage in the handling of this information, the compressed image may be decompressed to reconstruct the original image or an approximation of it. Image compression techniques play a crucial role in numerous applications including televideo conferencing, remote sensing, document and medical imaging, facsimile transmission, among others.
There exist a number of error-free compression techniques with which one may exactly reconstruct the compressed image. These techniques typically provide compression ratios on the order of from 2:1 to 10:1. Lossy encoding approaches on the other hand achieve increased compression ratios but result, by definition, in some degradation of the reconstructed image. Depending on the data, a minimal tradeoff in accuracy may yield significant increases in compression ratios.
Antenna data compression refers to the compression of complex voltage data empirically derived using an antenna and source of RF radiation. The voltage data are collected in a controlled environment and are representative of the voltages one would expect to measure in a field application using the same or a substantially similar antenna. The data are a record of the phase and amplitude of the complex voltages generated at an antenna array when irradiated with an emitter of known power, frequency and polarization. The relative orientation of the array and emitter are varied to acquire a sampling of data representative of the field of view of the antenna. Preferably, the data are characterized by a high signal-to-noise ratio at each of one or more distinct frequencies falling within a predetermined bandwidth. The voltage data are essential in passive, direction-finding systems that are used by ground-based radar and airborne vehicles. Angles-of-arrival are two body-relative polar angles representing the direction of the incoming wave. In the absence of a physical model, the voltage data are required for angle-of-arrival determinations of incoming electromagnetic signals when using search techniques relying on comparisons between measured and reference voltages.
In the present application specifically, the compression of the voltage data refers to the process by which the voltage data are mathematically modeled for later reconstruction in a fast and efficient manner using minimal computer storage memory. With no compression, the amount of memory required to adequately represent the reference voltage data with sufficient accuracy and resolution exceeds the practical memory and data transfer capacity of many volumetrically constrained applications such as with modern missiles and highly portable systems.
In missile and sensor pod applications, the problem of compression is exasperated by the complexity of the voltage data acquired by means of conformal antenna elements. Individual antenna elements are commonly mounted to, or in proximity of, the metallic or dielectric body of a missile, sensor pod or sensor suite housing. The presence of the body may partially shadow the antenna elements in addition to perturbing the field pattern in ways that are difficult to represent and predict with practical accuracy. As a result, both the phase and amplitude of antenna data may deviate substantially from those of simple models of sub-element antennas, such as the infinitesimal dipole model.
A second problem associated with voltage data acquired from antennas in missile applications arises from the fact that it is often necessary to place antenna elements within the constraints of special orientations as required for practical configurations with non-planar surfaces. The antenna elements are typically made to conform to the cylindrical or conical shape of the missile or sensor housing, giving rise to an array of elements having a diversity of polarization orientations. In the previous generation of missiles, arrays were comprised of receiving elements with substantially identical polarization angles resulting in an almost complete absence of polarization diversity. Each of the antenna elements possessed an equivalent polarization mismatch with the incoming wave giving rise to interferometric measurements that are a function of the difference in path length traveled by the incoming wave-front as it impinges upon the particular antenna element pair alone. The phase difference between pairs of antennas in a polarization-diverse array includes additional contributions because of the polarization mismatch between receiving antennas produced by the mounting orientations. The voltage received at each antenna is therefore a relatively complicated function of the angles of arrival and of the polarization of the incoming wave.
There are currently numerous transforms with which one may represent signal data. Among the most notable and commonly applied are the Fourier transform or Discrete Fast Fourier Transform (DFFT), as well as the Cosine transform. These transforms permit one to efficiently reconstruct a given signal exactly where: (1) the original signal is band-limited and (2) the data are acquired at a rate sufficiently high to satisfy the Nyquist sampling criterion. Where the signal data possess high frequency content, a large number of DFFT expansion terms must be retained in order to maintain the integrity of the original data. For purposes of antenna data compression, the Fourier transform is unsuitable for the reason that the voltage data are high frequency data. In the present application, for a typical quality, the Fourier and Cosine transforms provide unacceptably low compression ratios of four to one over the original data.
The primary object of the present invention is to provide a method for accurately representing the voltage data signals of an antenna system comprising a plurality of RF sensing elements with minimal data and minimal loss of accuracy. The complex voltage data signals are represented by the amplitude and phase of the received RF signal as measured by the plurality of sensing elements having differing and substantially known orientations of polarization. The voltage is primarily dependent on the two angles-of-arrival, two polarization parameters specifying the polarization of the incoming electromagnetic wave, and the frequency of the received wave.
This and other objects are achieved by representing each RF sensing element of an array of elements as the superposition of a plurality of xe2x80x9csub-element antennasxe2x80x9d with variable positions and orientations. The sub-element antennas are not physical antennas, but are mathematical representations of infinitesimal antennas distributed in free space. The sub-element antennas are preferably infinitesimal linear dipoles, but may be generalized to represent elements of elliptical polarization.
As physically realizable elements, the RF sensing elements are used in airborne vehicles to receive RF radiation propagating from one or more emitters. These receiving elements, when working in cooperation, comprise an array for performing direction-finding. Each receiving element has associated with it a distribution of sub-element antennas for modeling the voltage measured by the receiving element. Each of the sub-element antennas, or dipoles, is assigned a position in proximity to the associated receiving element. Each dipole is further assigned an initial polarization orientation preferably patterned after the receiving element. A hitherto unknown multiplying factor is then determined for each dipole. The factors constitute complex weight which, when determined, yield an expression for the voltage of a superposition of weighted dipoles that substantially and accurately represents the voltage detected by the receiving element for a variety of RF signal directions, frequencies, and polarizations.
The complex weights may be determined for each of the dipoles in accordance with the teachings of this invention. Below is described a method for solving for these weights in terms of the voltage data from which a system of linear equations is composed. The system of equations relates the voltage of the distribution of dipoles of unknown weights with the measured voltages acquired by means of actual, individual, receiving elements, each at discrete angles-of-arrival, frequencies, and polarizations. The system of equations is then operationally inverted and Maximum Likelihood (ML) or least squares techniques applied to determine the weights.
Given a distribution of weighted dipoles of known position and weighting, a reference voltage at a particular receiving element may be analytically reconstructed for an incoming electromagnetic wave (for arbitrary angles-of-arrival and polarization) from the sum of the voltages of the dipoles of the corresponding distribution. The accuracy of the reconstructed voltage may be reproduced with arbitrary precision for angles-of-arrival and frequencies that lie within the field of view and bandwidth as determined by the measured data. For a comparable loss of fidelity, the amount of data necessary to represent the distribution of dipoles is substantially smaller than that needed to store either the original voltage data or the data as compressed using other transformation techniques.
Referring now to FIGS. 1 and 2, where like reference numerals indicate like elements throughout the several views, the conformal antenna element array configuration used in deriving the sub-element antennas for data compression is shown. FIG. 1 illustrates the orientation and mounting of a RF antenna array 111 used to explain the present invention. An individual RF sensing element 101, when working in cooperation with other similar elements mounted on a surface 102, becomes an integral part of a RF antenna array 111. The mounting surface 102 illustrated is the shape of a cone with a rounded apex and could also be the shape of a right cone, a cylinder or any other convex or even concave surface. A principal axis 103 passes through a point at the center of the conical base of 102 and a point at the tip of the rounded apex. The antenna array 111 forms a ring around the principal axis 103. Coinciding with the principal axis 103 is the positive z-axis of a Cartesian coordinate system with its origin located at the point of intersection of the principal axis 103 and the plane containing the phase centers of the antenna elements comprising the antenna array 111. With the principal axis defining a forward horizontal directional axis, the y-axis 104 is a down directional axis and the x-axis 105 is a xe2x80x9cto the rightxe2x80x9d directional axis.
An individual sensing element 114 possesses a polarization orientation given by vector 112, which is in the plane created by the principal axis 103 and the phase center of the element 114. The angle between vector 112 and the plane including the axes 104 and 105 is the tilt angle 113 given by xcfx84, or tau. The tilt angle 113 is substantially the same value for each of the sensing elements comprising the array 111.
A vector directly from the center of the RF antenna array 111 to the RF emitting source 106 is designated the line-of-sight vector 107 and is the directional vector of the RF emitting source 106 relative to the RF antenna array 111. The angle created by the sides 103 and 107 is 109. The angle 109 is the first of two angles-of-arrival, xcex8, or theta.
Referring to FIG. 2, a view of the conformal antenna array 111 configuration is shown looking down the principal axis 103. In this configuration, a number of substantially similar, linearly polarized, antenna elements are located about a circular ring. The position of each antenna is defined in the plane containing the antenna elements by polar angle xcex1, or alpha, 208 as shown in FIG. 2. The angle created by the body-fixed, xe2x80x9cto the rightxe2x80x9d directional axis reference 105 and the line-of-sight vector projection 210 is the second angle-of-arrival 211, xcfx86, or phi.
The instantaneous electrical field of the incoming wave from a RF emitting source 106 contained in a plane perpendicular to the direction of the incoming wave is {overscore (E)}. In an arbitrary orthonormal coordinate system given by the triad of unit vectors {circumflex over ("xgr")}, {circumflex over (xcex7)}, and {circumflex over (xcex6)}, the electrical field of an incoming plane wave having emitter polarization angles xcex3, or gamma, and xcex4, or delta, propagating in the xcex6 direction is expressed by
{overscore (E)}=E"xgr" cos(xcfx89t){circumflex over ("xgr")}+Excex7 sin(xcfx89t+xcex4){circumflex over (xcex7)},xe2x80x83xe2x80x83[1]
where xcex4 is the phase by which the xcex7-component leads the "xgr"-component, xcfx89, or omega, is the oscillatory frequency of the electrical field and the angle xcex3 describes the relative values of E"xgr" and Excex7 as follows:
xcex3=arctan(Excex7/E"xgr") for 90 xe2x89xa7xcex3xe2x89xa70 degrees.xe2x80x83xe2x80x83[2]
The polarization of a receiving element, by definition, is the polarization of a wave radiated by the antenna in a given direction in the far field. In an arbitrary reference frame, the polarization of a wave emitted by such an antenna propagating in the {circumflex over (xcex6)} direction is given by
{circumflex over (p)}R=cos xcex3R{circumflex over ("xgr")}+sin xcex3R exp(jxcex4R){circumflex over (xcex7)}xe2x80x83xe2x80x83[3]
While the polarization state variables xcex4R and xcex3R completely define the polarization of the antenna in a given direction, they generally vary as a function of the particular xcex6 direction. Where reference is made to the polarization of an antenna without specifying the angular dependence, the polarization of a receiving antenna is understood to be the polarization of the antenna in the direction of the main beam.
The reference voltage is a function of the two states of polarization given by two pairs of angles, (xcex3, xcex4) and (xcex3R, xcex4R), as well as the direction of propagation of the incoming wave with respect to the orientation of the receiving element. The formulation of the voltage begins with the representation of the polarization of an elliptically polarized sub-element antenna located at the origin of the "xgr"R-xcex7R-xcex6R coordinate system:
{circumflex over (p)}R=cos xcex3R{circumflex over ("xgr")}R+sin xcex3R exp(jxcex4R){circumflex over (xcex7)}R,xe2x80x83xe2x80x83[4]
where the polarization vector lies in the plane perpendicular to the vector, {circumflex over (xcex6)}R. The xcex6R-axis then coincides with the direction that yields maximal power when the incoming wave has the same polarization as the receiving antenna in that same direction. In the "xgr"R-xcex7R-xcex6R coordinate system, the vector {circumflex over (xcex6)} is defined in the body-fixed coordinate system of the array mounting 102 by the angles, xcex8=xcex8R and xcfx86=xcfx86R The representation of the remaining antennas is identical, except for the rotation of the angular position and polarization orientation about the principal axis 103 of the array mounting 102.
The open circuit voltage, VOC, at a particular sensing element, is given by the inner product of the complex conjugate of the polarization state vector, which characterizes the state of the receiving antenna, and the electric field. The open circuit voltage is then given by
VOC={circumflex over (p)}*Rxc2x7E.xe2x80x83xe2x80x83[5]
Neglecting the field magnitude and the harmonic time dependence, the instantaneous complex voltage measured by an element at angular position xcex1 is proportional to voltage, v(xcex1), and may be expressed in the form:                                           v            ⁡                          (              α              )                                =                                    [                                                A                  ⁢                                      xe2x80x83                                    ⁢                                      cos                    ⁡                                          (                                              φ                        -                                                  φ                          R                                                -                        α                                            )                                                                      +                                  B                  ⁢                                      xe2x80x83                                    ⁢                                      sin                    ⁡                                          (                                              φ                        -                                                  φ                          R                                                -                        α                                            )                                                                      +                Z                            ]                        xc3x97                          exp              ⁡                              [                                                      -                    j                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      π                      ⁢                                              xe2x80x83                                            ⁢                      R                                        λ                                    ⁢                                      (                                                                  sin                        ⁢                                                  xe2x80x83                                                ⁢                        θ                        ⁢                                                  xe2x80x83                                                ⁢                                                  cos                          ⁡                                                      (                                                          φ                              -                              α                                                        )                                                                                              +                                              z                        ⁢                                                  xe2x80x83                                                ⁢                                                  cos                          ⁡                                                      (                            θ                            )                                                                                                                )                                                  ]                                                    ,                            [        6        ]            
where R is the radius of the surface of curvature 102, xcex is the wavelength of the received electromagnetic signal, and the following substitutions are made:
A=xe2x88x92[cos xcex3 cos xcex3R cos xcex8 cos xcex8Rxe2x88x92sin xcex3 exp(jxcex4)sin xcex3R exp (xe2x88x92jxcex4R)[xe2x80x83xe2x80x837]
B=xe2x88x92[sin xcex3 cos xcex3R cos xcex8R exp(jxcex4)+cos xcex3 sin xcex3R cos xcex8 exp(xe2x88x92jxcex4R)]xe2x80x83xe2x80x838]
Z=xe2x88x92cos xcex3 cos xcex3R sin xcex8 sin xcex8R.xe2x80x83xe2x80x83[9]
A special case of interest is that of the linearly polarized dipole antenna with a tilt angle, xcex8R=xcfx84. Consider the dipole located on the surface 102 at the angular position xcex1=0 with xcfx86R=0. The polarization state variables are given by xcex3R=xcfx80 and xcex4R=0, yielding coefficients A, B, and Z given as:
A=cos xcex3 cos xcex8 cos xcfx84xe2x80x83xe2x80x83[10]
B=sin xcex3 cos xcfx84exp(jxcex4)xe2x80x83xe2x80x83[11]
Z=cos xcex3 sin xcex8 sin xcfx84.xe2x80x83xe2x80x83[12]
The coefficients A, B, and Z are functions of the hitherto unknown angles-of-arrival as well and the polarization of the emitter and receiver.
The voltage represented by Equations 6 and 10 through 12 above is that of a single infinitesimal dipole and can be used to describe many electrically-small antennas in free space. However, when antennas are placed on a metallic or dielectric surface, or are partially shadowed by the mounting body on which they are attached, the voltages may vary substantially from predictions based upon the mathematical model of a single dipole.
Below is described a new process involving representative antenna field patterns, specifically a method by which voltage data may be compactly encoded and stored in memory prior to its decompression and application in angle-of-arrival determination as one of its immediate applications.
Voltage of Distribution of Dipole Antennas
A generalized model of the antenna field pattern may be obtained by representing the voltage received at each receiving element in terms of the voltages produced by a weighted distribution of infinitesimal dipoles such as those described in Equation 6. The description will be limited to the special case of the tilted linear dipoles where A, B, and Z are given by Equations 10 through 12. The generalized voltage of a group of such dipoles distributed about three positional degrees of freedom, xcex1, R, and z, and the single polarization tilt degree of freedom, xcfx84, is given as follows:                                                                         V                ⁡                                  (                                      θ                    ,                    φ                    ,                    γ                    ,                    δ                    ,                    λ                                    )                                            =                              xe2x80x83                            ⁢                              ∫                                  ∫                                      ∫                                          [                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                          γ                          ⁢                                                      xe2x80x83                                                    ⁢                          cos                          ⁢                                                      xe2x80x83                                                    ⁢                          θ                          ⁢                                                      xe2x80x83                                                    ⁢                                                      cos                            ⁡                                                          (                                                              φ                                -                                α                                                            )                                                                                                      +                                                                                                                                                                                                      xe2x80x83                                ⁢                                                      sin                    ⁢                                          xe2x80x83                                        ⁢                    γ                    ⁢                                          xe2x80x83                                        ⁢                                          exp                      ⁡                                              (                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          δ                                                )                                                              ⁢                                          sin                      ⁡                                              (                                                  φ                          -                          α                                                )                                                                              ]                                            xc3x97                              exp                [                                  j                  ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                            λ                                        [                                          R                      ⁢                                              xe2x80x83                                            ⁢                      sin                      ⁢                                              xe2x80x83                                            ⁢                      θ                      ⁢                                              xe2x80x83                                            ⁢                                              cos                        (                                                  φ                          -                                                                                                                                                                                                                                                                                                                          xe2x80x83                                                ⁢                                                  α                          )                                                                    +                                              z                        ⁢                                                  xe2x80x83                                                ⁢                        cos                        ⁢                                                  xe2x80x83                                                ⁢                        θ                                                              ]                                    ]                                ⁢                                  ⅆ                  α                                ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  R                                ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  z                                ⁢                                  ∫                                                            w                      ⁡                                              (                                                  α                          ,                          R                          ,                          z                          ,                          τ                                                )                                                              ⁢                    cos                    ⁢                                          xe2x80x83                                        ⁢                    τ                    ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      τ                                                                                  +                                                                                          xe2x80x83                            ⁢                              cos                ⁢                                  xe2x80x83                                ⁢                γ                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                θ                ⁢                                  ∫                                      ∫                                          ∫                                              exp                        [                                                  j                          ⁢                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              π                                                        λ                                                    [                                                                                    R                              ⁢                                                              xe2x80x83                                                            ⁢                              sin                              ⁢                                                              xe2x80x83                                                            ⁢                              θ                              ⁢                                                              xe2x80x83                                                            ⁢                                                              cos                                ⁡                                                                  (                                                                      φ                                    -                                    α                                                                    )                                                                                                                      +                                                                                                                                                                                                                                                                            xe2x80x83                                    ⁢                                      z                    ⁢                                          xe2x80x83                                        ⁢                    cos                    ⁢                                          xe2x80x83                                        ⁢                    θ                                    ]                                ]                            ⁢                              ⅆ                α                            ⁢                              xe2x80x83                            ⁢                              ⅆ                R                            ⁢                              ⅆ                z                            ⁢                              ∫                                                      w                    ⁡                                          (                                              α                        ,                        R                        ,                        z                        ,                        τ                                            )                                                        ⁢                                      sin                    ⁡                                          (                      τ                      )                                                        ⁢                                                            ⅆ                      τ                                        .                                                                                                          [        13        ]            
By defining two new weighting functions, given by
g(xcex1, R, z)=∫w(xcex1, R, z, xcfx84)cos xcfx84dxcfx84xe2x80x83xe2x80x83[14]
h(xcex1, R, z)=∫w(xcex1, R, z, xcfx84)sin xcfx84dxcfx84,xe2x80x83xe2x80x83[15]
the voltage of Equation 13 may be expressed as                                                                         V                ⁡                                  (                                      θ                    ,                    φ                    ,                    γ                    ,                    δ                    ,                    λ                                    )                                            =                              xe2x80x83                            ⁢                              ∫                                  ∫                                      ∫                                                                  g                        ⁡                                                  (                                                      α                            ,                            R                            ,                            z                                                    )                                                                    [                                                                        cos                          ⁢                                                      xe2x80x83                                                    ⁢                          γ                          ⁢                                                      xe2x80x83                                                    ⁢                          cos                          ⁢                                                      xe2x80x83                                                    ⁢                          θ                          ⁢                                                      xe2x80x83                                                    ⁢                                                      cos                            ⁡                                                          (                                                              φ                                -                                α                                                            )                                                                                                      +                                                                                                                                                                                                      xe2x80x83                                ⁢                                  sin                  ⁢                                      xe2x80x83                                    ⁢                  γ                  ⁢                                      xe2x80x83                                    ⁢                                      exp                    ⁡                                          (                                              j                        ⁢                                                  xe2x80x83                                                ⁢                        δ                                            )                                                        ⁢                                      sin                    ⁡                                          (                                              φ                        -                        α                                            )                                                                      ]                            xc3x97                              exp                [                                  j                  ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                            λ                                        [                                          R                      ⁢                                              xe2x80x83                                            ⁢                      sin                      ⁢                                              xe2x80x83                                            ⁢                      θ                      ⁢                                              xe2x80x83                                            ⁢                                              cos                        (                                                  φ                          -                                                                                                                                                                                                                                                                                                                          xe2x80x83                                                ⁢                                                  α                          )                                                                    +                                              z                        ⁢                                                  xe2x80x83                                                ⁢                        cos                        ⁢                                                  xe2x80x83                                                ⁢                        θ                                                              ]                                    ]                                ⁢                                  ⅆ                  α                                ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  R                                ⁢                                  ⅆ                  z                                            +                                                                                          xe2x80x83                            ⁢                              cos                ⁢                                  xe2x80x83                                ⁢                γ                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                θ                ⁢                                  ∫                                      ∫                                          ∫                                                                        h                          ⁡                                                      (                                                          α                              ,                              R                              ,                              z                                                        )                                                                          ⁢                                                  exp                          [                                                      j                            ⁢                                                                                          2                                ⁢                                                                  xe2x80x83                                                                ⁢                                π                                                            λ                                                        [                                                          R                              ⁢                                                              xe2x80x83                                                            ⁢                              sin                              ⁢                                                              xe2x80x83                                                            ⁢                              θ                              ⁢                                                              xe2x80x83                                                            ⁢                                                              cos                                (                                                                  φ                                  -                                                                                                                                                                                                                                                                                                                                                                                                                    xe2x80x83                                            ⁢                                              α                        )                                                              +                                          z                      ⁢                                              xe2x80x83                                            ⁢                      cos                      ⁢                                              xe2x80x83                                            ⁢                      θ                                                        ]                                ]                            ⁢                              ⅆ                α                            ⁢                              xe2x80x83                            ⁢                              ⅆ                R                            ⁢                                                ⅆ                  z                                .                                                                        [        16        ]            
If one further presumes that each of the dipoles of the distribution are restricted to discrete values for the position, the voltage is expressed as the finite triple sum:                                           V            ⁡                          (                              θ                ,                φ                ,                γ                ,                δ                ,                λ                            )                                =                                                    ∑                                  n                  =                  1                                N                            ⁢                                                ∑                                      m                    =                    1                                    M                                ⁢                                                      ∑                                          k                      =                      1                                        K                                    ⁢                                                                                    g                        nmk                                            ⁡                                              [                                                                              cos                            ⁢                                                          xe2x80x83                                                        ⁢                            γ                            ⁢                                                          xe2x80x83                                                        ⁢                            cos                            ⁢                                                          xe2x80x83                                                        ⁢                            θ                            ⁢                                                          xe2x80x83                                                        ⁢                                                          cos                              ⁡                                                              (                                                                  φ                                  -                                                                      α                                    n                                                                                                  )                                                                                                              +                                                      sin                            ⁢                                                          xe2x80x83                                                        ⁢                            γ                            ⁢                                                          xe2x80x83                                                        ⁢                                                          exp                              ⁡                                                              (                                                                  j                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  δ                                                                )                                                                                      ⁢                                                          sin                              ⁡                                                              (                                                                  φ                                  -                                                                      α                                    n                                                                                                  )                                                                                                                                    ]                                                              xc3x97                                          exp                      ⁡                                              [                                                  j                          ⁢                                                                                                                    2                                ⁢                                π                                                            λ                                                        ⁡                                                          [                                                                                                                                    R                                    m                                                                    ⁢                                  sin                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  θ                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      cos                                    ⁡                                                                          (                                                                              φ                                        -                                                                                  α                                          n                                                                                                                    )                                                                                                                                      +                                                                                                      z                                    k                                                                    ⁢                                  cos                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  θ                                                                                            ]                                                                                                      ]                                                                                                                  +                          cos              ⁢                              xe2x80x83                            ⁢              γ              ⁢                              xe2x80x83                            ⁢              sin              ⁢                              xe2x80x83                            ⁢              θ              ⁢                                                ∑                                      n                    =                    1                                    N                                ⁢                                                      ∑                                          m                      =                      1                                        M                                    ⁢                                                            ∑                                              k                        =                        1                                            K                                        ⁢                                                                  h                        nmk                                            ⁢                                              exp                        ⁡                                                  [                                                      j                            ⁢                                                                                                                            2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  π                                                                λ                                                            ⁡                                                              [                                                                                                                                            R                                      m                                                                        ⁢                                    sin                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    θ                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          cos                                      ⁡                                                                              (                                                                                  φ                                          -                                                                                      α                                            n                                                                                                                          )                                                                                                                                              +                                                                                                            z                                      k                                                                        ⁢                                    cos                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    θ                                                                                                  ]                                                                                                              ]                                                                                                                                                        ,                            [        17        ]            
where xcex1n represents the N discrete values of the position angle, xcex1, Rm represents the M values of the position variable, R, and zk represents the K values of the position along the principal axis 103.
Determination of Weights
The weights of the dipoles are then determined from the original voltage data empirically acquired. The complex voltage data are presumed to have been acquired over the field of view, and vary with the angles-of-arrival, xcex8i (i=0, 1, . . . , I) and xcfx86j (j=0, 1, . . . , J), in increments of approximately one to several degrees. The voltage data should also be acquired as functions of the polarization states, xcex3c and xcex4d, of the incoming RF wave, as well as wavelength, xcexf . An individual datum, V(xcex8i, xcfx86j, xcex3c, xcex4d, xcexf), therefore corresponds to one element of a set of Ixc3x97Jxc3x97Cxc3x97Dxc3x97F equations in the weights gnmk and hnmk given by                               V          ⁡                      (                                          θ                i                            ,                              φ                j                            ,                              γ                c                            ,                              δ                d                            ,                              λ                f                                      )                          =                                            ∑                              n                =                1                            N                        ⁢                                          ∑                                  m                  =                  1                                M                            ⁢                                                ∑                                      k                    =                    1                                    K                                ⁢                                                      g                    nmk                                    [                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      γ                    c                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    i                                    ⁢                                      cos                    ⁡                                          (                                                                        φ                          j                                                -                                                  α                          n                                                                    )                                                                                                    +                      sin            ⁢                          xe2x80x83                        ⁢                          γ              c                        ⁢                          exp              ⁡                              (                                  j                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    d                                                  )                                      ⁢                          sin              ⁡                              (                                                      φ                    j                                    -                                      α                    n                                                  )                                      ⁢                          "AutoLeftMatch"              ]                        xc3x97                          exp              ⁡                              [                                  j                  ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                                                    λ                        f                                                              ⁡                                          [                                                                                                    R                            m                                                    ⁢                          sin                          ⁢                                                      xe2x80x83                                                    ⁢                                                      θ                            i                                                    ⁢                                                      cos                            ⁡                                                          (                                                                                                φ                                  j                                                                -                                                                  α                                  n                                                                                            )                                                                                                      +                                                                              z                            k                                                    ⁢                          cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      θ                            i                                                                                              ]                                                                      ]                                              +                      cos            ⁢                          xe2x80x83                        ⁢                          γ              c                        ⁢            sin            ⁢                          xe2x80x83                        ⁢                          θ              i                        ⁢                                          ∑                                  n                  =                  1                                N                            ⁢                                                ∑                                      m                    =                    1                                    M                                ⁢                                                      ∑                                          k                      =                      1                                        K                                    ⁢                                                            h                      nmk                                        ⁢                    exp                    [                    j                    ⁢                                                                                                                        2                            ⁢                                                          xe2x80x83                                                        ⁢                            π                                                                                λ                            f                                                                          [                                                                                                            R                              m                                                        ⁢                            sin                            ⁢                                                          xe2x80x83                                                        ⁢                                                          θ                              i                                                        ⁢                            cos                            (                                                          φ                              j                                                                                -                                                                                    α                              n                                                        )                                                    +                                                                                                                    z                                k                                                            ⁢                              cos                              ⁢                                                              xe2x80x83                                                            ⁢                                                              θ                                i                                                                                      ]                                                                          ]                                            .                                                                                                                              [        18        ]            
Letting the set of observations (xcex8i, xcfx86j, xcex3c, xcex4d, xcexf) be collectively represented by "THgr"xcexc and the triplet (xcex1n, Rm, zk) with a single index variable, xcex2v, Equation 18 may now be written as                                           V            ⁡                          (                              Θ                μ                            )                                =                                                    ∑                                  v                  =                  1                                                  N                  ·                  M                  ·                  K                                            ⁢                                                G                  v                                ⁢                                  Ξ                  ⁡                                      (                                                                  Θ                        μ                                            ;                                              β                        v                                                              )                                                                        +                                          ∑                                  v                  =                  1                                                  N                  ·                  M                  ·                  K                                            ⁢                                                H                  v                                ⁢                                  Ψ                  ⁡                                      (                                                                  Θ                        μ                                            ;                                              β                        v                                                              )                                                                                      ,                            [        19        ]            
where                                           Ξ            ⁡                          (                                                Θ                  μ                                ;                                  β                  v                                            )                                =                                    {                                                cos                  ⁢                                      xe2x80x83                                    ⁢                                      γ                    c                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      θ                    i                                    ⁢                                      cos                    ⁡                                          (                                                                        φ                          j                                                -                                                  α                          n                                                                    )                                                                      +                                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      γ                    c                                    ⁢                                      exp                    ⁡                                          (                                              jδ                        d                                            )                                                        ⁢                                      sin                    ⁡                                          (                                                                        φ                          j                                                -                                                  α                          n                                                                    )                                                                                  }                        xc3x97                          exp              ⁡                              [                                  j                  ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                                                    λ                        f                                                              ⁡                                          [                                                                                                    R                            m                                                    ⁢                          sin                          ⁢                                                      xe2x80x83                                                    ⁢                                                      θ                            i                                                    ⁢                                                      cos                            ⁡                                                          (                                                                                                φ                                  j                                                                -                                                                  α                                  n                                                                                            )                                                                                                      +                                                                              z                            k                                                    ⁢                          cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      θ                            i                                                                                              ]                                                                      ]                                                    ⁢                  
                ⁢        and                            [        20        ]                                          Ψ          ⁡                      (                                          Θ                μ                            ;                              β                v                                      )                          =                  cos          ⁢                      xe2x80x83                    ⁢                      γ            c                    ⁢          sin          ⁢                      xe2x80x83                    ⁢                      θ            i                    ⁢                                    exp              ⁡                              [                                  j                  ⁢                                                                                    2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                                                    λ                        f                                                              ⁡                                          [                                                                                                    R                            m                                                    ⁢                          sin                          ⁢                                                      xe2x80x83                                                    ⁢                                                      θ                            i                                                    ⁢                                                      cos                            ⁡                                                          (                                                                                                φ                                  j                                                                -                                                                  α                                  n                                                                                            )                                                                                                      +                                                                              z                            k                                                    ⁢                          cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      θ                            i                                                                                              ]                                                                      ]                                      .                                              [        21        ]            
Using matrix and vector notations, the analytical reference voltage may be expressed as
V=MW,xe2x80x83xe2x80x83[22]
where
M=["Xgr"xcexa8]xe2x80x83xe2x80x83[23]
and
WT=[G, H].xe2x80x83xe2x80x83[24]
The voltage of Equation 22 is then equated to the set of measured complex voltages, VMEAS, the number of components being given by the product, Uxe2x80x2=Ixc3x97Jxc3x97Cxc3x97Dxc3x97F. In the special case where the number of measured voltages is equal to the number of weights, the solution is given by
WT=Mxe2x88x921VMEAS.xe2x80x83xe2x80x83[25]
In general, the matrix M is a rectangular matrix having a number of rows given by Uxe2x80x2, and a number of columns given by Uxe2x80x3=2Nxc3x97Mxc3x97K, where Uxe2x80x2xe2x89xa7Uxe2x80x3. A psuedo-inverse solution for the weights is obtained using a least-square fit technique as one technique among others commonly applied and obvious to skilled practitioners.
Where the matrix M is singular or near-singular, the weight vector is determined using Singular Value Decomposition (SVD). In accordance with the SVD technique, the singular value decomposition of M begins with the creation of a column-orthogonal matrix P of size Uxe2x80x2 by Uxe2x80x3, a diagonal matrix S (=diag└s1, s2 . . . sN┘) of size Uxe2x80x3 by Uxe2x80x3 with positive or zero elements, and the complex conjugate transpose (indicated by xe2x80xa0) of the orthogonal matrix Q (i.e., Qxe2x80xa0) of the size Uxe2x80x3 by Uxe2x80x3, such that
M=PSQxe2x80xa0.xe2x80x83xe2x80x83[26]
The next step is to determine a threshold, T, at which s1xe2x89xa7xcex7 where t=1, 2, . . . , T. By partitioning the matrices P, S and Q as follows:                               P          =                      [                                                                                                      P                      T                                        1                                                                                                              P                                                                        U                          xe2x80x3                                                -                        T                                                              2                                                                        ]                          ,                            [        27        ]                                          Q          =                      [                                                                                                      Q                      T                                        1                                                                                                              Q                                                                        U                          xe2x80x3                                                -                        T                                                              2                                                                        ]                          ,                  
                ⁢        and                            [        28        ]                                S        =                              [                                                                                S                    1                                                                    0                                                                              0                                                                      S                    2                                                                        ]                                              U              xe2x80x3                        -            T                    T                                    [        29        ]            
the weight vector is calculated using the following matrix operations:
W=(Q1S1xe2x88x921P1xe2x80xa0)VMEAS.xe2x80x83xe2x80x83[30]
With the weights of the distribution of infinitesimal dipoles determined, Equation 18 now yields a mathematical expression for the complex voltage as determined by a receiving element of the antenna system. The expression may now be used as a generalized antenna model by which the voltage may be determined for any arbitrary angle-of-arrival within the field of view for which the weights where optimized.
The present invention may also be used as a method for determining the angles-of-arrival where the voltage data have previously been acquired for the frequency of interest. The reference voltages are reconstructed for each antenna element at each increment of the polar angles xcex8 and xcfx86 of the field of view. The true direction angles are then determined by minimizing the difference between actual voltage measurements and the references voltages calculated using the present method.